A Line-Search-Free Coordinate Proximal Predictor-Corrector Method for Monotone Absolute Value Equations

Abstract

We consider the absolute value equation (AVE) $Ax-|x|=b$ under residual monotonicity. We characterize this property through the symmetric part of the coefficient matrix, thereby allowing nonsymmetry, and propose a coordinate proximal predictor-corrector (CPPC) method based on an AVE-specific forward-proximal decomposition. An exact scalar proximal update on the largest proximal-residual coordinate generates a predictor point, while a positive-alignment identity certifies the ensuing full-residual separating-hyperplane correction without backtracking. With the current matrix product cached, each iteration requires one new full matrix-vector product. For a nonempty solution set, we prove Fejér monotonicity, whole-sequence convergence, and an $O(K^{-1/2})$ best-iterate residual bound. A positive monotonicity margin further ensures unique solvability for every right-hand side and global linear convergence. Numerical results identify regimes in which the reduced per-iteration work yields shorter solution times.